A uniqueness theorem concerning gravity fields

It is well known that a uniform spherical shell produces the same gravitational field outside itself as that of a concentric uniform solid sphere with the same total mass. A given external field may therefore be produced by a variety of mass systems even under the restriction that the bodies of the system should have a prescribed uniform density. It can be asked what further conditions on the system are sufficient to ensure its uniqueness. The problem is of interest in connexion with the geological interpretation of local anomalies in the earth's gravitational field and work has already been done on it by several authors, (3), (6), (7), (8), (9). The results proved so far have all been restricted to systems consisting of a single body and the purpose of the present paper is to extend one of these results to include more general mass systems. The example of the spherical shell, mentioned above, might suggest that the absence of cavities in the bodies is a sufficient additional condition to ensure uniqueness. This is not so however. Celmiņš (3) gives an example of two different mass systems which have the same external field, each system consisting of three bodies of the same uniform density without any cavities in them.